Factoring formulas
Below are some factoring formulas that are used to factor some common math expressions
a
^{2} - b
^{2} = ( a - b ) × ( a + b )
a
^{4} - b
^{4} = ( a - b ) × ( a + b ) × ( a
^{2} + b
^{2} )
a
^{6} - b
^{6} = ( a - b ) × ( a + b ) × ( a
^{2} - ab + b
^{2} ) × ( a
^{2} + ab + b
^{2} )
a
^{8} - b
^{8} = ( a - b ) × ( a + b ) × ( a
^{2} + b
^{2} ) × ( a
^{4} + b
^{4} )
a
^{3} + b
^{3} = ( a + b ) × ( a
^{2} - ab + b
^{2} )
a
^{3} - b
^{3} = ( a - b ) × ( a
^{2} + ab + b
^{2} )
a
^{5} - b
^{5} = ( a - b ) × ( a
^{4} + a
^{3}b + a
^{2}b
^{2} + ab
^{3} + b
^{4} )
a
^{5} + b
^{5} = ( a + b ) × ( a
^{4} - a
^{3}b + a
^{2}b
^{2} - ab
^{3} + b
^{4} )
a
^{6} - b
^{6} = ( a - b ) × ( a
^{5} + a
^{4}b + a
^{3}b
^{2} + a
^{2}b
^{3} + ab
^{4} + b
^{5} )
a
^{6} + b
^{6} = ( a
^{2} + b
^{2} ) × ( a
^{4} - a
^{2}b
^{2} + b
^{4} )
a
^{7} - b
^{7} = ( a - b ) × ( a
^{6} + a
^{5}b + a
^{4}b
^{2} + a
^{3}b
^{3} + a
^{2}b
^{4} + ab
^{5} + b
^{6} )
a
^{4} + a
^{2}b
^{2} + b
^{4} = ( a
^{2} + ab + b
^{2} )× ( a
^{2} - ab + b
^{2} )
a
^{4} + 4b
^{4} = ( a
^{2} + 2ab + 2b
^{2} )× ( a
^{2} - 2ab + 2b
^{2} )
Trick to factor a
^{n} - b
^{n} when n is an odd number. You cannot use this trick if n is even or to factor a
^{n} + b
^{n}
First start by writing ( a - b ) × ( ..................................................)
Then, fill in the parenthesis on the right
To do this, follow this guideline.
Subtract 1 from n. For example, if n = 7 as in a
^{7} - b
^{7}, subtract 1 from 7 to get 6.
The first term is always going to be the first variable that is a raised to the power of 6
The first term is always going to be the second variable that is b raised to the power of 6
The operation inside is always a plus
So it is going to look like ( a - b ) × (a
^{6} + .................................................. + b
^{6})
Now here is the real tricky part
To get the next term, is a
^{5}b. This is done by subtracting 1 from 6 and incorporating the other variable
Now all you have to do is to keep subtracting 1 to the exponent of a and adding 1 to the exponent of b as shown below
The next term will be a
^{4}b
^{2}.
Do this until the variable a disappears and you will end up with the answer already shown above
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